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F -theory GUTs

J. Marsano

Enrico Fermi Institute

University of Chicago

In collaboration with: N. Saulina, S. Schäfer-Nameki

0808.1286

,0808.2450

,0904.3932

,0906.4672

,0912.0272

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Particle Physics and String Theory

• How can we hope to say anything about particle physics giventhe complexity of string vacua?

• A hint from nature that we are in a good situation. . .

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Particle Physics and String Theory

• How can we hope to say anything about particle physics giventhe complexity of string vacua?

• A hint from nature that we are in a good situation. . .

• Apparent unification ofcouplings in MSSM atM GUT ∼ 1016 GeV 

→ M SUSY ≪ M GUT ≪ M Planck

14 15 16 17LogM 1 GeV

24

26

28

30

32

1Αr

t

• Suggests that particle physics captured by SUSY gauge theorythat isn’t too sensitive to details of quantum gravity

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Bottom-up Approach

[Aldazabal, Ibanez, Quevedo, Uranga], [Gray, He, Jejjala, Nelson][Verlinde, Wijnholt]

• Look for framework with natural separation between gauge andgravity degrees of freedom

→ Type II models with branes

1. Smooth compactification with intersecting branes (this talk)

[Cvetic, Shiu, Uranga], + many others

2. Branes probing singular compactifications

[Aldazabal, Ibanez, Queveco, Uranga], [Verlinde, Wijnholt], + many others

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Charged Matter

SU(5)

U(1)

SU(6) Bifundamental • Charged matter from openstrings with one end on thestack

• Other end on some other

D-brane, orientifold plane,→ "Matter branes"

• Can be described by larger rank gauge theory with nontrivialadjoint vev

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Worldvolume Gauge Theory

• Study worldvolume gauge theory with gauge group G ⊃ SU (5)

• Nontrivial vev for adjoint scalar field

φAdj = 0 → G → SU (5) {×U (1)m }

• Large brane rotations → string scale vevs

• Nevertheless can compute holomorphic data

• Spectrum• Superpotential

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MSSM from SU (5) GUT

• Supersymmetric SU (5)GUT GUT

3×10M  ∼

Q  ∼ (3, 2)+1/6

U c  ∼ (3, 1)−2/3

E c  ∼ (1, 1)+1

3×5M  ∼

D c  ∼ (3, 1)+1/3

L ∼ (1, 2)−1/2

5H  ∼

H u  ∼ (1, 2)+1/2

H (3)u  ∼ (3, 1)−1/3

5H  ∼

H d  ∼ (1, 2)−1/2

H (3)d  ∼ (3, 1)+1/3

W ⊃ λUP10M × 10M × 5H , λDOWN10M × 5M × 5H 

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Which G ’s are Possible?

G → SU (5)×U (1)

k Adj(G ) → Adj(SU (5))⊕

Adj(U (1)k 

⊕ [Bifundamentals]

• MSSM matter

10 and 5 ∈ Bifundamentals 

• MSSM Superpotential

10× 10× 5, 10× 5× 5 ⊂ Adj(G )3

=⇒ G ⊃ E 7

Forced to nonperturbative realm of type II theories

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F-theory and M-theory

M -Theory Models

• Intersecting "6-brane" models

• 7-dimensional gauge theory

on a R 3-manifold X 3[Pantev, Wijnholt]

• Charged matter localized atpoints in X 3

• Yukawas exponentiallysuppressed but no candidatemodel of flavor

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F-theory and M-theory

M -Theory Models

• Intersecting "6-brane" models

• 7-dimensional gauge theory

on a R 3-manifold X 3[Pantev, Wijnholt]

• Charged matter localized atpoints in X 3

• Yukawas exponentiallysuppressed but no candidatemodel of flavor

F -Theory Models

• Intersecting "7-brane" models

• GUT → 8-dimensional gauge

theory on a C surface S [Donagi, Wijnholt]

[Beasley, Heckman, Vafa]

• Charged matter localized oncurves in S 

• Some proposed mechanismsfor flavor structure

[Heckman, Vafa]

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F-theory and M-theory

M -Theory Models

• Intersecting "6-brane" models

• 7-dimensional gauge theory

on a R 3-manifold X 3[Pantev, Wijnholt]

• Charged matter localized atpoints in X 3

• Yukawas exponentiallysuppressed but no candidatemodel of flavor

• Difficult to study

F -Theory Models

• Intersecting "7-brane" models

• GUT → 8-dimensional gaugetheory on a C surface S 

[Donagi, Wijnholt][Beasley, Heckman, Vafa]

• Charged matter localized oncurves in S 

• Some proposed mechanismsfor flavor structure

[Heckman, Vafa]

• Simpler to study

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Focus

• We will study GUTs in F -theory, namely

8-dimensional E 8 gauge theory on a C surface S 

• Will now see how much progress we can make towards

• Breaking GUT Group

• Suppressing Proton Decay

• Addressing Unification Issues

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Aside: Connection to Heterotic

[Vafa], [Morrison, Vafa], [Friedman, Morgan, Witten][Curio, Donagi], [Donagi, Wijnholt], [Hayashi, Tatar, Toda, Watari, Yamazaki]

• 8d E 8 gauge theory → SU (5) with φadj = 0

• Field configuration for φAdj must satisfy certain BPS equations

• Solutions can be constructed using spectral cover

• Near GUT 7-branes, geometry a local K3-fibration over S 

• Can embed into global K3 fibration with a Heterotic dual

• Geometric deformation↔ E 8 spectral bundle

• Constructed from spectral cover

Gauge Theory Description ↔ "Heterotic Description"

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Charged Matter

SU(5)

U(1)

SU(6) Bifundamental

SU(5)

S

5

"Matter Curve"

GUT

Charged matter is effectively 6-dimensional

• Spectrum of 4d multiplets requires further dimensional reduction• # of 4d multiplets can be adjusted with worldvolume fluxes

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GUT-Breaking

SU (5) → SU (3) × SU (2)×U (1)Y 

24 → (8, 1)0 ⊕ (1, 3)0 ⊕ (1, 1)0 ⊕

(3, 2)−5/6 ⊕ cc

• Must break SU (5) and project out the lepto-quarks• Promising idea: Internal U (1)Y  flux on S 

[Beasley, Heckman, Vafa] [Donagi, Wijnholt]

• Can lift lepto-quarks as well as Higgs triplets

H ∼

H u  ∼ (1,2)+1/2

H (3)u  ∼ (3, 1)

−1/3

ffH ∼

H d  ∼ (1,2)

−1/2

H (3)d  ∼ (3,1)+1/3

ff

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Dimension 4 Proton Decay

• Must forbid the dangerous operator

W Proton Decay ∼ 10M × 5M × 5M 

• How to do it?

• Discrete symmetries – Difficult in practice[Tatar, Tsuchiya, Watari]

• Continuous symmetries – plentiful[JM, Saulina, Schäfer-Nameki]

E 8 → SU (5)× U (1)4

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Which Symmetries?

W MSSM  ∼ 10M × 10M × 5H  + 10M × 5M × 5H 

+10M × 5M × 5M  + 10M × 5H × 5H 

• MSSM Superpotential Preserves a 1-parameter family of U (1)’s

10M  5M  5H  5H 

U (1) 1 q − 3 −2 2− q 

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Which Symmetries?

W MSSM  ∼ 10M × 10M × 5H  + 10M × 5M × 5H 

+10M × 5M × 5M  + 10M × 5H × 5H 

• MSSM Superpotential Preserves a 1-parameter family of U (1)’s

10M  5M  5H  5H 

U (1) 1 q − 3 −2 2− q 

• When q = 0, Q ∼ Q Y  + Q B −L

• Gauged B − L models

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Which Symmetries?

W MSSM  ∼ 10M × 10M × 5H  + 10M × 5M × 5H 

+10M × 5M × 5M  + 10M × 5H × 5H 

• MSSM Superpotential Preserves a 1-parameter family of U (1)’s

10M  5M  5H  5H 

U (1) 1 q − 3 −2 2− q 

• When q = 0, Q ∼ Q Y  + Q B −L

• Gauged B − L models• q = 0 → PQ symmetry

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Dimension 5 Proton Decay

• There are tight bounds on the operator responsible for dimension

5 proton decay1

M 

 d 2θQ 3L ⊂

1

M 

 d 2θ

10M × 10M × 10M × 5M 

• Mechanisms like "missing partner" have been proposed in

F -theory models before Beasley, Heckman, Vafa

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Dimension 5 Proton Decay

• There are tight bounds on the operator responsible for dimension5 proton decay

1

M 

 d 2θQ 3L ⊂

1

M 

 d 2θ

10M × 10M × 10M × 5M 

• Mechanisms like "missing partner" have been proposed in

F -theory models before Beasley, Heckman, Vafa

• Unfortunately, KK modes will generate any allowed operatorsJM, Saulina, Schäfer-Nameki

10M  5M  5H  5H 

U (1) 1 q − 3 −2 2− q 

• The above operator carries U (1) charge q 

→ Allowed if q = 0

=⇒ We must realize U (1) with q = 0 – need U (1)PQ  symmetry

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Comments on U (1)PQ  Symmetries

• Prevent a bare µ term

W µ ∼ µ 5H × 5H 

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Comments on U (1)PQ  Symmetries

• Prevent a bare µ term

W µ ∼ µ 5H × 5H 

• If a PQ -charged field X  carries a SUSY-breaking vev, a small µ

term can be generated via Giudice-Masiero type mechanismfrom  

d 4θ1

M X † × 5H × 5H 

• Possible solution for µ/B µ problem in gauge mediation[Ibe, Kitano], [JM, Saulina, Schäfer-Nameki]

• PQ gauge boson gives additional contributions to SUSY-breakingsoft masses

[Heckman, Vafa]

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Getting the Right Symmetries

• Our U (1)’s come from the parent gauge group,

E 8 ⊃ SU (5)GUT × SU (5)⊥

E 8 → SU (5) × U (1)4 φ ∼

t 1 0 0 0 00 t 2 0 0 00 0 t 3 0 0

0 0 0 t 4 00 0 0 0 t 5

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Getting the Right Symmetries

• Our U (1)’s come from the parent gauge group,

E 8 ⊃ SU (5)GUT × SU (5)⊥

E 8 → SU (5) × U (1)4 φ ∼

t 1 0 0 0 00 t 2 0 0 0

0 0 t 3 0 0

0 0 0 t 4 00 0 0 0 t 5

• φAdj varies over internal space

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Getting the Right Symmetries

• Our U (1)’s come from the parent gauge group,

E 8 ⊃ SU (5)GUT × SU (5)⊥

E 8 → SU (5) × U (1)4 φ ∼

t 1 0 0 0 00 t 2 0 0 00 0 t 3 0 0

0 0 0 t 4 0

0 0 0 0 t 5

• φAdj varies over internal space

• Only gauge invariant quantities, such as trm φAdj, need bewell-defined

→ t i ’s are permuted by a monodromy group G Monod as we move alongthe internal space, S 

• G Monod effectively quotients the theory, removing most (or all)

U (1)’s

• G Monod intrinsically connected to "global" structure on S 

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Implications of PQ Symmetries

[JM, Saulina, Schäfer-Nameki]

• Sadly, the rosy scenario we have described doesn’t like PQ symmetries very much

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Implications of PQ Symmetries

[JM, Saulina, Schäfer-Nameki]

• Sadly, the rosy scenario we have described doesn’t like PQ symmetries very much

• Whenever we have a PQ symmetry and

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Implications of PQ Symmetries

[JM, Saulina, Schäfer-Nameki]

• Sadly, the rosy scenario we have described doesn’t like PQ symmetries very much

• Whenever we have a PQ symmetry and

1. All 3 generations of MSSM matter localized on a single curve→ for flavor structure

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Implications of PQ Symmetries

[JM, Saulina, Schäfer-Nameki]

• Sadly, the rosy scenario we have described doesn’t like PQ symmetries very much

• Whenever we have a PQ symmetry and

1. All 3 generations of MSSM matter localized on a single curve→ for flavor structure

2. GUT-breaking via U (1)Y  flux

we also get

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Implications of PQ Symmetries

[JM, Saulina, Schäfer-Nameki]

• Sadly, the rosy scenario we have described doesn’t like PQ symmetries very much

• Whenever we have a PQ symmetry and

1. All 3 generations of MSSM matter localized on a single curve→ for flavor structure

2. GUT-breaking via U (1)Y  flux

we also get

3. Extra light non-GUT exotics

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PQ -Symmetry =⇒ Exotic Particles

E ti ?!?!?!?!?

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Exotics?!?!?!?!?

• What to do about exotics?1. Find some other way to get flavor structure

[Ibanez, Font], [Dudas, Palti]

2. Use a different mechanism to break the GUT group

3. Look for exotic structures in string theory beyond E 8

4. Lift the exotics through coupling to a PQ -charged singlet, X 

Z d 2θXf Exoticf Exotic

• Reminiscent of gauge mediation

• Interesting phenomenology?

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Exotics are not the only problem with these models

U ifi ti P bl i M d l ith U(1) Fl

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Unification Problem in Models with U (1)Y  Flux

• 8-dimensional gauge theory has an important 1-loop divergence

[Donagi, Wijnholt], [Wijnholt]

ln(Λ)

 trAdj (F ∧ F ∧ F ∧ F )

• Represents a "local tadpole" that must be canceled globally in a

consistent string compactification[Conlon], [Conlon, Palti]

• Background F Y  leads to distortions of 4-d coupling at the KK scale[Blumenhagen], [Wijnholt]

θZ R3,1

F ∧ F → θZ R3,1

F ∧ F + ln„

ΛM KK 

«Z R3,1

F ∧ F Z S 

F Y  ∧ F Y 

• Non-universal shifts – splits gauge couplings at the high scale

Comparison with Observation

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Comparison with Observation• Worry about couplings at M Z  – this is what we actually measure

α−1i  (M Z ) = αGUT −

β (MSSM )i 

2πlnM KK 

M z 

− β (thresh)i 

2πln

ΛM KK 

• Splittings α−1i  (M Z )− α−1

 j  (M Z ) independent of αGUT butsensititive to threshold contributions• MSSM alone agrees with experiment to within 0.5%

• With correction, we sit on threshold of acceptibility

• Most rosy estimates for Λ can lead to ∼ 0.

5% but most are somewhatlarger

Comparison with Observation

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Comparison with Observation• Worry about couplings at M Z  – this is what we actually measure

α−1i  (M Z ) = αGUT −

β (MSSM )i 

2πln

M KK 

M z 

−

β (thresh)i 

2πln

Λ

M KK 

−β (exotic)

2πln

M KK M exotic

• Splittings α−1i  (M Z )− α−1

 j  (M Z ) independent of αGUT butsensititive to threshold contributions

• MSSM alone agrees with experiment to within 0.5%

• With correction, we sit on threshold of acceptibility

• Most rosy estimates for Λ can lead to ∼ 0.

5% but most are somewhatlarger

• Contributions from exotics open up a parameter space of betteragreement but we need

M exotic

> 1013−14 GeV 

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Our two problems may be able to "cancel" each other

Moving Beyond the "Charged Sector"

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Moving Beyond the Charged Sector

• Crucial to understand singlet fields that are not localized on theSU (5) branes

1. Right-handed neutrinos are SU (5) singlets

2. Exotic masses come from coupling to SU (5) singlets3. Potential SUSY-breaking spurions?

• Singlets probe geometry away from GUT brane surface, S 

• Gauge theory perspective not useful

• Cannot rely so strongly on Heterotic input

Global Fluxes

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Global Fluxes

First problem: Fluxes

• Fluxes that induce chirality constructed essentially throughHeterotic techniques

• Global extension of these fluxes beyond the local neighborhood of

S has always been assumed

• Need intrinsic F -theory understanding of global fluxes in order tostart moving away from the GUT 7-branes

• Some ideas for this. . .

[JM, Saulina, Schäfer-Nameki, in progress]

Compact Models

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Compact Models

• Several compact models do exist (modulo flux assumptions)[JM, Saulina, Schäfer-Nameki] [Blumenhagen, Grimm, Jurke, Krause, Weigand]

• Toy models at best

• Moduli stabilization not addressed

• In known examples, D3-brane tadpole doesn’t leave much room for

extra fluxes for this

N D 3 =χ(Y 4)

24−

1

2

Z Y 4

G ∧G 

Z Y 4

G ∧ G ≥ 0

Summary

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Summary

• Unification suggests particle physics embedded in

nonperturbative regime of type II

• "Charged sector" described by 7-d or 8-d gauge theory

• Local geometry has dual Heterotic description that provides

important input

• Structure of 8-d gauge theory (F -theory models) is very rigid

• Serious implications for phenomenology in very large class of

string models

• Moving away from 8-d gauge theory requires new technical

advancements

• A study of 7-d gauge theories relevant for M -theory modelswould be interesting to undertake